The key comes from Budaen below the central tower
The “Tower Key” for the doors above the Storm Gem is the Forbidden Archives Key. Find the lever below the upper staircase, rotate the stairs, climb their new route and take the central tower elevator down. Speak to the injured knight Budaen for the key, then return to the upper locked doors. The Storm Gem itself does not unlock them.

On this page
Find the lever below the upper staircase
Start: the huge upper staircase near Knowledge Keeper Wourha's study. Its top leads to the Storm Gem pedestal; the nearby upper doors remain locked until you get Budaen's key.

- Return below the Storm Gem platform and look for the lower opening or landing beneath the staircase. Use Wind Dash to reach the landing with the mechanism.
- If you cannot see the landing, picture the stairs rotating 180 degrees. Search where their bottom end would arrive after that turn. The lever is at that side of the lower stair area.
- Approach the lever and use Interact to rotate the staircase. Follow the repositioned stairs into the newly reachable section.

The gap below the stairs is the easily missed branch. Returning to the same upper locked doors without changing the staircase leaves you at the same dead end.
Take the elevator down to Budaen
- Climb the rotated staircase and follow its new connection into the central tower room with the elevator.
- Ride the elevator down. Continue into the room with the injured, seated knight pictured in the answer above.
- Speak to Budaen and continue his dialogue until he gives you the key. He explains that the book called to him and asks you to take the key to the forbidden archives.
Done when: you receive the Forbidden Archives Key. Reaching Budaen's room without talking to him does not complete the task.
Use the key at the upper Archives doors
Retrace the elevator and stair route to the two locked doors near the upper Storm Gem route. Interact with a door to use the Forbidden Archives Key and enter the Archives. A Crystal Monument lies beyond the entrance; save before continuing deeper.
If a door is still locked, first confirm that you finished Budaen's conversation and that you are back at the upper Archives doors. The island's Endu Shrine Gate Key opens a different gate.
This page starts after you have reached the upper tower. Opening the tower's outside entrance and solving its earlier balcony puzzle are separate tasks.